If \(\frac{d-2}{2}\le k<\frac{d}{2}\) , then the second condition in (REF ) is vacuously true provided that \(1<p,q<\infty \)  and \(\frac{1}{p}-\frac{1}{q}=\frac{2k}{d}\) . Hence, in this case, the inequality (REF ) holds for all \(1<p,q<\infty \)  satisfying the gap condition \(\frac{1}{p}-\frac{1}{q}=\frac{2k}{d}\) . If \(k<\frac{d-2}{2}\) , however, the second condition in (REF ) is nontrivial.See, for example, Figures REF  and REF  in Section , where the necessary and sufficient condition for (REF ) is described by the thick line segments. If \(k\ge \frac{d}{2}\) , then (REF ) is not true because the set (REF ) is empty. Theorem REF  extends the aforementioned results in [1]} and [2]} for \(k=1\)  and in [3]} for \(q=p^{\prime }\) .
